Look-ahead schemes for block Toeplitz systems and formal orthogonal matrix polynomials
Marc Van Barel, Adhemar Bultheel · 1997
In this paper, a so-called auxiliary matrix polynomial Xn(z) and a true right formal orthogonal matrix polynomial (FOMP) A 1 n (z) is connected to each well-conditioned leading principal block submatrix of a given block Toeplitz matrix. From these two matrix polynomials, all other right FOMPs of block n of a system of block biorthogonal matrix polynomials with respect to the block Toeplitz moment matrix can be computed in an efficient way. The two matrix polynomials Xn(z) and A 1 n (z) also represent all solutions of a homogeneous interpolation problem connected to the symbol of the block Toeplitz moment matrix. On the other hand, they form part of the parameters of an inversion formula for the connected block Toeplitz matrix. However, numerical experiments show loss of accuracy when this inversion formula is used. Instead the solution of the block Toeplitz system can be updated leading to a weakly stable algorithm using O( 2 ) FLOPS if is the dimension of the block Toeplitz mat...